A continuous function is defined on the interval

MATHEMATICS · Multi-part · CBSE · Mathematics · Integrals · Integrals

Problem

A continuous function

f

is defined on the interval

[0,4]

. The following definite integrals are known:

\begin{aligned} \text{(1)} \int_{0}^{1} f(x)\,dx &= 2,\\ \text{(2)} \int_{1}^{2} f(x)\,dx &= 4,\\ \text{(3)} \int_{2}^{4} f(x)\,dx &= 9. \end{aligned}

Answer the following sub-questions, showing each step clearly. (a) Part 1: (i) Find

\displaystyle \int_{0}^{2} f(x)\,dx

. State the property of definite integrals you use. (b) Part 2: (ii) Hence obtain

\displaystyle \int_{1}^{4} f(x)\,dx

. Show all working. (c) Part 3: (iii) Using your result from part (ii), evaluate

\displaystyle \int_{4}^{1} f(x)\,dx

. Give a brief reason for the sign of your answer.